Scientific journal
European Journal of Natural History
ISSN 2073-4972

A VERY INEXPENSIVE SCHEME ON RFEM TO USE IN CFD AND OTHER PROBLEMS

Mohammad Reza Akhavan Khaleghi 1
1 The Office of Counseling and Research Fluid Engineering and Aerodynamic
Note, Reference [1] should be read before reading this paper. In this paper I am going to use the Reduced Finite Element Method (RFEM), see [1] with a constant amount of Element Degree Limiter, Field Variable Limiter (EDL, FVL) coefficient to obtain an inexpensive scheme of non-oscillatory, for this purpose I used third degree Lagrangian elements on full upwind difference scheme (FUDS) and its result was very successful.
Reduced Finite Element Method
Non-oscillatory
Full upwind difference scheme
Third-order scheme
1. Khaleghi M.R.A. A new computational package for using in CFD and other problems.

The non-oscillatory shape functions [1] is written as follows:

moh01.wmf (1)

Where moh02.wmf is Element Degree Limiter, Field Variable Limiter (EDL, FVL), moh03.wmf is p-order shape functions (in this paper degree of p is 3) and moh04.wmf, moh05.wmf and moh06.wmf are reference functions I used equation (40) of [1] as reference function that is

moh07.wmf moh08.wmf moh09.wmf (2)

moham1.wmf

Fig. 1. 1D advective equation moh16.wmf

 

moham2.wmf

 

Fig. 2. 1D convection-diffusion equation moh17.wmf

 

moham3.wmf

Fig. 3. Grid for flow over a NACA 0012 airfoil

moham4.wmf

Fig. 4. Pressure distribution on surface of NACA 0012 airfoil, steady-state

moham5.wmf

Fig. 5. Grid for flow around a cylinder.

moham6.wmf

Fig. 6. Flow around a cylinder, steady-state

Where moh10.wmf and moh11.wmf are liner shape functions, by putting equation (2) in (1) and also moh12.wmf we have

moh13.wmf (3)

Equation (3) is non-oscillatory shape functions that will be used in this paper for CFD problems.

Examples

In this section, I am going to use the equation (3) as shape function for approximating a few equation and compare its result with non-constant moh14.wmf. As first example, I approximated 1D advective equation by Point Collocation weight function, see figure (1). In second example, I approximated 1D convection-diffusion equation in moh15.wmf by Galerkin weight function, see figure (2). In third example, I approximated 2D Euler equations for pressure distribution on surface of NACA 0012 airfoil [1] by Galerkin weight function and infinite elements for non-solid boundaries, see figure (4), and flow around a cylinder [1] by Subdomain Collocation weight function see figure (6).

Conclusions

As can be seen from the results, with any number of elements solutions are non-oscillatory, and when we use fine mesh (usually in CFD is used) solutions are very close together. So, using of constant EDL, FVL is useful.